Relativized circuit complexity

Non-cancellative Boolean circuits: A generalization of monotone boolean circuits

CHAPTER 2 - A Catalog of Complexity Classes

We study the robustness of complete sets for various complexity classes. A complete set A is robust if for any f(n)-dense set S /spl isin/ P, A - S is still complete, where f(n) ranges from log(n), polynomial, to subexponential. We show that robustness can be used to separate complexity classes: For every /spl les//sub m//sup p/-complete set A for EXP and any subexponential dense sets S /spl isin/ P, A - S is still Turing complete and under a reasonable hardness assumption even /spl les//sub m//sup p/-complete. For EXP and the delta levels of the exponential hierarchy we show that for every Turing complete set A and any log-dense set S /spl isin/ P, A - S is still Turing complete. There exists a 3-truth-table complete set A for EEXPSPACE, and a log-dense set S /spl isin/ P such that A - S is not Turing complete. This implies that settling this issue for EEXP will either separate P from PSPACE or PUfrom EXP. We show that the robustness results for EXP and the delta levels of the exponential hierarchy do not relativize.

In this chapter, we introduce the definitions of automata Turing machines. We present different flavors of Turing machines including deterministic, probabilistic, and multi-tape machines. We present several complexity classes, i.e. collections of problems that can be solved by a machine with similar resources, including the P and NP classes. We present the Church-Turing thesis, which states that if a problem can be solved by a physical machine, then it can also be solved by any deterministic Turing machine. In the last part of the chapter, we introduce quantum Turing machines and the quantum circuit model. We study the complexity class BQP. This is the class of problems that can be solved by a quantum computer with polynomial resources and a small error probability. Finally, we introduce QMA, the quantum version of the complexity class NP. We present the k-LOCAL problem, and we show that this problem is in QMA

Recently, related to the open problem of whether deterministic and nondeterministic space (especially lower-level) complexity classes are separated, the inkdot Turing machine was introduced. An inkdot machine is a conventional Turing machine capable of dropping an inkdot on a given input tape for a landmark, but not to pick it up nor further erase it. In this paper, we introduce a finite state version of the inkdot machine as a weak recognizer of the properties of digital pictures, rather than a Turing machine supplied with a one-dimensional working tape. We first investigate the sufficient spaces of three-way Turing machines to simulate two-dimensional inkdot finite automaton, as preliminary results. Next, we investigate the basic properties of two-dimensional inkdot automaton, i.e. the hierarchy based on the number of inkdots and the relationship of two-dimensional inkdot automata to other conventional two-dimensional automata. Finally, we investigate the recognizability of connected pictures of two-dimensional inkdot finite machines.

Let C denote one of the complexity classes “polynomial time,” “logspace,” or “nondeterministic logspace.” We introduce a logic L (C) inv and show generalizations and variants of the equivalence ( L (C) inv captures C if and only if there is an almost C-optimal algorithm in C for the set Taut of tautologies of propositional logic). These statements are also equivalent to the existence of a listing of subsets in C of Taut by corresponding Turing machines and equivalent to the fact that a certain parameterized halting problem is in the parameterized complexity class XC uni .

Domino games which describe computations of alternating Turing machines in the same way as dominoes (tiling systems) encode computations of deterministic and nondeterministic Turing machines are considered. The domino games are two-person games in the course of which the players build up domino tilings of a square of prescribed size. Acceptance of an alternating Turing machine corresponds to a winning strategy for one player—the number of moves in the game is the number of alternations of the Turing machine. Let $\operatorname{ATIME}(T(n), m)$ denote the class of all sets that are accepted by some alternating Turing machine in time $T(n)$ with at most m alternations. It is shown that any problem in such a complexity class can be reduced to the strategy problem for some domino game. In particular domino games which are complete in the classes $\Sigma_{m}^{p}$, and $\Pi_{m}^{p}$, of the polynomial time hierarchy are found. This corresponds to the approach of Savelsbergh and van Emde Boas and of Lewis and Papadimitriou, who have shown that the theory of NP-completeness may also be founded on a finite domino problem instead of the satisfiability problem for propositional formulae. Similar generalizations are possible for domino connectability problems: Domino thread games where the players build up a thread of dominoes are introduced; the first player wins if he can ultimately connect two given points. It is shown that a certain class of such games captures deterministic time complexity. In particular, games are found whose strategy problems are P-complete. In the last section applications to the complexity of simple prefix classes in logical theories are briefly discussed.

Many variants of P systems with active membranes are able to solve traditionally intractable problems. Sometimes they also characterize well known complexity classes, depending upon the computational features they use. In this paper we continue the investigation of the importance of (minimal) cooperative rules to increase the computational power of P systems. In particular, we prove that monodirectional shallow chargeless P systems with active membranes and minimal cooperation working in polynomial time precisely characterise $$\mathbf{P }^{\#{\mathbf{P }}}_{\parallel }$$ , the complexity class of problems solved in polynomial time by deterministic Turing machines with a polynomial number of parallel queries to an oracle for a counting problem.

On the Circuit Complexity of Sigmoid Feedforward Neural Networks

THE INTERFACE BETWEEN LANGUAGE THEORY AND COMPLEXITY THEORY

A generalized Grzegorczyk hierarchy and low complexity classes

Observations on complete sets between linear time and polynomial time

We examine the power of nondeterministic finite automata with acceptance of an input word defined by a so called leaf language, i. e., a condition on the sequence of leaves in the automaton's computation tree. We study leaf languages either taken from one of the classes of the Chomsky hierarchy, or taken from a time- or space-bounded complexity class. We contrast the obtained results with those known for leaf languages for Turing machines and Boolean circuits.

This paper introduces a two-dimensional alternating Turing machine (2-ATM) which is an extension of an alternating Turing machine to two-dimensions. This paper also introduces a three-way two-dimensional alternating Turing machine (TR2-ATM) which is an alternating version of a three-way two-dimensional Turing machine. We first investigate a relationship between the accepting powers of space-bounded 2-ATM's (or TR2-ATM's) and ordinary space-bounded two-dimensional Turing machines (or three-way two-dimensional Turing machines). We then introduce a simple, natural complexity measure for 2-ATM's (or TR2-ATM's), called -&-ldquo;leaf-size-&-rdquo;, and provides a spectrum of complexity classes based on leaf-size bounded computations. We finally investigate the recognizability of connected patterns by 2-ATM's (or TR2-ATM's).

Two-dimensional alternating turing machines